
For the single tensor model, we prove some general results for the \((m,n)\) -Rényi reflected entropy, including various continuity arguments as a function of \(m\)  which establishes independence of \(m\)  away from the phase transition. These will serve as checks on our later computations, and help motivate the prescription we use to analytically continue in \((m,n)\) . We then describe the phase diagram of dominant saddle points at integer \(m/2,n\) . See sec:ineq and sec:phasenm.

In the single tensor model, we use a Schwinger-Dyson re-summation method similar to [1]}, [2]} to compute the reflected spectrum. We make an assumption about the class of diagrams that dominates, and prove this to be correct in various limits. The reflected spectrum thus obtained has two main components: A single eigenvalue representing the disconnected EW phase, and a broad peak with \(\exp (2 EW(A:B))\)  eigenvalues representing the connected EW phase. These two contributions are always present irrespective of the relevant phase, but their respective weights move around as we tune the bond dimensions in the tensor network. They make dominant contributions to the reflected entropy in their respective phases. We show a cartoon of the reflected spectrum for the single tensor model in fig:spectrum1. See sec:EE and sec:SD.

Based on the above we argue that near the phase transition both sets of eigenvalues contribute. In the single tensor, the reflected entropy takes the form:
\(S_R(A:B) = -p_0\ln p_0 -p_1\ln p_1 + p_1 \left( \ln \chi ^2_A - \frac{\chi ^2_A}{2\chi _B^2} \right) + O(\chi _{A,B}^{-2})\) 
where \(\chi _X \gg 1\)  are the bond dimensions for \(X=A,B,C\)  and the classical probabilities are:
\(p_0 = \frac{\chi _C}{\chi _A\chi _B} \;_2F_1 (1/2,-1/2;2; \frac{\chi _C}{\chi _A \chi _B})^2\,, \qquad p_1 + p_0 = 1,\) 
We argue that these results can be understood as arising from an effective description of the entanglement structure for the canonically purified state as a superposition of tensor networks states. Here, one state is described by a doubled random tensor network, representing the connected phase, while the other tensor network is trivially factorized:
\(\includegraphics [width=0.8]{fig/CP_effective_1}\) 
The two terms are weighted by the associated probabilities \(p_0\)  and \(p_1 = 1 - p_0\) . The reduced density matrices on \(AA^\star \)  are approximately orthogonal, so these represent (approximate) superselection sectorsThese are also sometimes referred to as \(\alpha \) -blocks or \(\alpha \) -sectors. with an associated area operator:
\(\mathcal {L}_{AA^\star } = \bigoplus _{s=0,1} S_{vN}(\zeta _s)\) 
where \(\zeta _1 \approx 1_{AA^\star }/\chi _A^2\)  and \(\zeta _0 = \left| 1_{A}/\chi _A \right> \left< 1_{A}/\chi _A \right|\) .
We also check the above results against numerics. See sec:spectrum and sec:num.

For the hyperbolic networks we compute \(S_R^{(m,n)}(A:B)\)  at large bond dimension \(\chi \)  and at integer \(m/2,n\)  by looking for dominant saddle points in the effective spin model that computes the Haar random average. There are potentially many competing saddle points, so this turns out to be a difficult problem. However when the entanglement wedge is in the connected phase we conjecture, by analogy to the single tensor, that two types of saddles give the main contribution for all \(n,m/2\) . These phases were not accounted for in the original discussion of Ref. [3]}. However, the difference to the naive approach does not show up in the reflected entropy \(S_R\) . The \((m,n)\) -Rényi reflected entropy in the connected phase (\(x>1/2\) ) can be written as:
\(S_R^{(m,n)}(A:B) = {\rm min} \left.{\left\lbrace \begin{array}{ll}\frac{n}{n-1} I(A,B)\end{array}\\\ln \chi \times \min _{0 \le \mathcal {A} \le \mathcal {A}_x}\left[2 \mathcal {A} + \frac{4 n}{n-1} \ln \right.\cosh (\mathcal {A}_x- \mathcal {A}) \right]}\right.\) 
}

where \(\mathcal {A}_x =\ln \frac{1 + \sqrt{1-x}}{\sqrt{x}}\)  and \(I(A,B) =2 \ln \chi \ln (1-x)/x\)  is the mutual information.
The first term arises from a disconnected saddle, that is still present in the connected phase, while the second term represents the connected saddles. The area \(\mathcal {A}_x\)  represents the cross-sectional area of the \(AB\)  entanglement wedge. While \(\mathcal {A}\)  represents the cross-sectional area of a “pinched” entanglement wedge which arises due to the non-trivial tension of the entanglement brane, when \(n > 1\) .
We can again give an interpretation of this result using a superposition of networks, analogous to eq:CPeffective1TN:
\(\includegraphics [width=0.8]{fig/CP_effective_2.pdf}\) 
where the state described by each side of this equation have the same \(S_R^{(m,n)}\) . The tensor networks on the right hand side are doubled and glued networks similar to those discussed in [4]}. The probabilities above can be read directly from ansrsr and we will further expound upon the meaning of these pictures in sec:generalTN.

Our results will resolve the order of limits issue pointed out in Ref. [5]}. In particular if one applies the naive LM-like saddle point analysis in the limit \((m,n) \rightarrow (1,1)\)  then the “wrong” phase can dominate if one takes the limit \(m \rightarrow 1\)  first and then \(n \rightarrow 1\) , leading to a formula different from eq:srequals2ew. This later formula fails several quantum information bounds, so one can rule out this order of limits on these grounds. However a better understanding of this issue and justification of ordermn is desired. In fact we will show that no such order of limits issue occurs after we give a prescription for continuing the \((m,n)\) -Rényi reflected entropies from \(m \ge 2\)  to \(m=1\) . We will give evidence for this prescription based on the single random tensor, and show how it is consistent with a more general set of QI bounds.

